Not any 4-manifold admits non-trivial solutions to (REF ), so half-integrability of \(K\) 
imposes restrictions. We may call equation (REF ) the “shear-free condition”, since in
Lorentzian general relativity its solutions define what are known as shear-free null geodesic congruences:
the real vector field \(\ell ^{a}=\varphi ^{A}\bar{\varphi }^{A^{\prime }}\)  is tangent to a null geodesic congruence which is
shear-free (see e.g. [1]} for the definition of shear of a geodesic congruence).
If, for example, one also asks the metric to be Einstein, then by the Goldberg-Sachs theorem the Weyl
tensor must be algebraically special, which is a restriction on the curvature.
Still, many physically interesting spacetimes in general relativity satisfy this property; in particular,
stationary black hole solutions such as Kerr or Schwarzschild (with or without cosmological constant).
